Negative magnetic eddy diffusivities from test-field method and multiscale stability theory

TL;DR

This paper compares the test-field method and multiscale stability theory for calculating magnetic eddy diffusivities, confirming their agreement and deriving new relations for flows with opposite small-scale velocities.

Contribution

It analytically demonstrates the equivalence of two approaches for magnetic eddy diffusivity evaluation and derives new relations for flows with opposite velocities.

Findings

Both methods yield fully consistent results.

Computed eddy diffusivity tensors confirm previous studies.

Derived relations between eddy diffusivity tensors for opposite flows.

Abstract

The generation of large-scale magnetic field in the kinematic regime in the absence of an alpha-effect is investigated by following two different approaches, namely the test-field method and multiscale stability theory relying on the homogenisation technique. We show analytically that the former, applied for the evaluation of magnetic eddy diffusivities, yields results that fully agree with the latter. Our computations of the magnetic eddy diffusivity tensor for the specific instances of the parity-invariant flow-IV of G.O. Roberts and the modified Taylor-Green flow in a suitable range of parameter values confirm the findings of previous studies, and also explain some of their apparent contradictions. The two flows have large symmetry groups; this is used to considerably simplify the eddy diffusivity tensor. Finally, a new analytic result is presented: upon expressing the eddy diffusivity tensor in terms of solutions to auxiliary problems for the adjoint operator, we derive relations between magnetic eddy diffusivity tensors that arise for opposite small-scale flows v(x) and -v(x).